On the connection between the Kannan-Lovasz-Simonovits conjecture and the variance conjecture for convex bodies and some of its algorithmic implications

January 28, 2013
Ronen Eldan | Weizmann Institute of Science

We consider the uniform measure over a high-dimensional isotropic convex body. We prove that, up to logarithmic factors, the corresponding isoperimetric minimizers are ellipsoids. We thus establish a connection between two well-known conjectures regarding the uniform measure over a high dimensional convex body, namely the Thin-Shell conjecture and the conjecture by Kannan-Lovasz-Simonovits (KLS), showing that a positive answer to the former will imply a positive answer to the latter (up to a logarithmic factor). Our proof relies on the analysis of the eigenvalues of a certain random-matrix-valued stochastic process related to a convex body. We also discuss some algorithmic implications of our methods.

Speaker Details

Ronen Eldan Received is B.A degree in Maths from the Open University of Israel in 2005, which was later extended to an additional discipline in Physics in 2006, at the Tel Aviv University. In 2012, He graduated his Ph.D. studies in mathematics at the Tel Aviv University, under the supervision of Prof. V. Milman and Prof. B. Klartag, specializing in probability, high dimensional convex geometry and computational geometry. Before and during most of his studies, he also worked as a computer security specialist, applied mathematician and team-leader of a programming group for the IDF and in the industry. He was an intern in the Theory Group at Microsoft research in Autumn 2011. Currently, he is a Post-Doctoral fellow at the department of Mathematics and Computer Science in the Weizmann Institute of Science.